A121598 -id:A121598 - cp's OEIS frontend

A121601 Decimal expansion of cosecant of 22.5 degrees = csc(Pi/8).

2, 6, 1, 3, 1, 2, 5, 9, 2, 9, 7, 5, 2, 7, 5, 3, 0, 5, 5, 7, 1, 3, 2, 8, 6, 3, 4, 6, 8, 5, 4, 3, 7, 4, 3, 0, 7, 1, 6, 7, 5, 2, 2, 3, 7, 6, 6, 9, 8, 5, 3, 9, 0, 5, 5, 0, 9, 7, 7, 9, 6, 7, 3, 3, 8, 1, 6, 1, 6, 2, 0, 8, 2, 9, 2, 2, 3, 8, 4, 1, 0, 1, 9, 0, 3, 7, 0, 7, 4, 4, 0, 3, 8, 5, 2, 5, 6, 2, 8, 6, 4, 9, 2, 7, 7

Offset: 1

Rick L. Shepherd, Aug 09 2006

1 + csc(Pi/8) is the radius of the smallest circle into which 9 unit circles can be packed ("r=3.613+ Proved by Pirl in 1969", according to the Friedman link, which has a diagram).
csc(Pi/8) is the distance between the center of the larger circle and the center of each unit circle that touches the larger circle.
A rectangle of length L and width W is a called a silver rectangle if L=rW, where r is the silver ratio; i.e., r = 1+sqrt(2). The diagonal has length D = sqrt(W^2+L^2), so that D/W = sqrt(4+2*sqrt(2)) = csc(Pi/8). - Clark Kimberling, Apr 04 2011
This algebraic integer of degree 4 also gives the length ratio diagonal/side of the longest diagonal in the regular octagon. The minimal polynomial is x^4  - 8*x + 8. In the power basis of Gal(Q(rho(8))/Q), with rho(8) = sqrt(2 + sqrt(2)) = A179260 it is -2*rho(8) + 1*rho(8)^3 which equals sqrt(2)*rho(8). - Wolfdieter Lang, Oct 28 2020

			2.6131259297527530557132863468543743071675223766985390550977...

D. Mumford et al., Indra's Pearls, Cambridge 2002; see p. 362. - N. J. A. Sloane, Nov 22 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 4

Cf. A121598, A179260.

SetDefaultRealField(RealField(100)); R:=RealField(); 1/Sin(Pi(R)/8); // G. C. Greubel, Nov 02 2018

evalf(1/sin(Pi/8),120); # Muniru A Asiru, Nov 02 2018

RealDigits[Csc[Pi/8],10,130][[1]] (* corrected by Harvey P. Dale, Jul 28 2012 *)

PARI
```
1/sin(Pi/8)
```

Equals 2*sqrt(2)*cos(Pi/8).

Equals Product_{k >= 0} (8*k + 4)^2/((8*k + 1)*(8*k + 7)). - Antonio Graciá Llorente, Mar 11 2024

A323601 Decimal expansion of sin(Pi/7).

Original entry on oeis.org

4, 3, 3, 8, 8, 3, 7, 3, 9, 1, 1, 7, 5, 5, 8, 1, 2, 0, 4, 7, 5, 7, 6, 8, 3, 3, 2, 8, 4, 8, 3, 5, 8, 7, 5, 4, 6, 0, 9, 9, 9, 0, 7, 2, 7, 7, 8, 7, 4, 5, 9, 8, 7, 6, 4, 4, 4, 5, 4, 7, 3, 0, 3, 5, 3, 2, 2, 0, 3, 2, 5, 1, 6, 5, 3, 1, 9, 8, 4, 2, 1, 5, 2, 0, 7, 8, 4, 0, 2, 1, 7, 7, 4, 4, 5, 6, 1, 0, 2, 0, 8, 8, 7, 4, 4, 1

Offset: 0

Vaclav Kotesovec, Jan 19 2019

			0.43388373911755812047576833284835875460999072778745987644454730353220325...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for algebraic numbers, degree 6.

Cf. A019829 (sin(Pi/9)), A232736 (sin(Pi/14)).

Cf. A121598, A272487.

SetDefaultRealField(RealField(100)); R:= RealField(); Sin(Pi(R)/7); // G. C. Greubel, Feb 08 2019

Mathematica
```
RealDigits[Sin[Pi/7], 10, 120][[1]]
```

default(realprecision, 100); sin(Pi/7) \\ G. C. Greubel, Feb 08 2019

polrootsreal(64*x^6-112*x^4+56*x^2-7)[4] \\ Charles R Greathouse IV, Feb 05 2025

numerical_approx(sin(pi/7), digits=100) # G. C. Greubel, Feb 08 2019

Root of the equation 64*x^6 - 112*x^4 + 56*x^2 - 7 = 0. (Other +- A232735 and +- 0.7818314... = +- cos(3*Pi/14))
Equals sqrt((196 + 7*i*2^(2/3)*(21*i*sqrt(3) - 7)^(1/3)*(i + sqrt(3)) + i*2^(4/3)*(21*i*sqrt(3) - 7)^(2/3)*(2*i + sqrt(3)))/336), where i is the imaginary unit.
Equals cos(5*Pi/14).
From Gleb Koloskov, Jul 15 2021: (Start)
Positive root of the equation x^3 + sqrt(7)/2*x^2 - sqrt(7)/8 = 0.
Equals ((4*sqrt(7)*(13+3*sqrt(3)*i))^(1/3)+28*(4*sqrt(7)*(13+3*sqrt(3)*i))^(-1/3)-2*sqrt(7))/12, where i is the imaginary unit. (End)
Equals 1/A121598 = A272487/2. - Hugo Pfoertner, Dec 15 2024
This^2 + A073052^2=1. - R. J. Mathar, Aug 31 2025

A374957 Decimal expansion of the circumradius of a regular heptagon with unit side length.

Original entry on oeis.org

1, 1, 5, 2, 3, 8, 2, 4, 3, 5, 4, 8, 1, 2, 4, 3, 2, 5, 2, 6, 2, 0, 5, 7, 5, 1, 1, 1, 7, 7, 3, 4, 2, 7, 5, 5, 6, 7, 2, 2, 2, 5, 0, 9, 4, 3, 8, 0, 3, 1, 6, 0, 5, 8, 1, 0, 3, 1, 5, 5, 3, 1, 4, 8, 2, 3, 3, 4, 2, 6, 6, 7, 1, 3, 8, 9, 2, 3, 9, 7, 9, 8, 1, 8, 9, 5, 5, 5, 7, 1

Offset: 1

Paolo Xausa, Jul 26 2024

			1.15238243548124325262057511177342755672225094380...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Regular polygon.
Index entries for algebraic numbers, degree 6.

Cf. A374971 (apothem), A374972 (sagitta), A178817 (area).
Cf. circumradius of other polygons with unit side length: A020760 (triangle), A010503 (square), A300074 (pentagon), A285871 (octagon), A375151 (9-gon), A001622 (10-gon), A375190 (11-gon), A188887 (12-gon).
Cf. A073052, A121598, A272487.

Mathematica
```
First[RealDigits[Csc[Pi/7]/2, 10, 100]]
```

 5/sin(Pi/7) \\ Charles R Greathouse IV, Feb 05 2025

Equals csc(Pi/7)/2 = A121598/2.
Equals 1/(2*sin(Pi/7)) = 1/A272487.
Equals A374971/cos(Pi/7) = A374971/A073052.
Largest of the 6 real-valued roots of 7*x^6-14*x^4+7*x^2-1=0. - R. J. Mathar, Aug 29 2025

A121603 Numbers n such that the radius of the smallest circle into which n unit circles can be packed is 1 + csc(Pi/k), where k >= 2 is an integer.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11

Offset: 1

Rick L. Shepherd, Aug 09 2006

nonn

Corresponding k values are in A121604. For these n, the centers of k unit circles can form a regular k-gon with sides of length 2 centered at the center of the larger circle. From the diagrams in the link it appears likely that 13,18,19 are the next three terms.

			See A121602 for the case n=11 involving a 9-gon.

E. Friedman, Erich's Packing Center: "Circles in Circles"

Cf. A121604, A121570, A121598, A121601, A121602.

A121604 Numbers k such that the radius of the smallest circle into which A121603(m) unit circles can be packed is 1 + csc(Pi/k).

Original entry on oeis.org

2, 3, 4, 5, 6, 6, 7, 8, 9

Offset: 0

Rick L. Shepherd, Aug 09 2006

nonn

From the diagrams in the link it appears likely that 10,12,12 are the next three terms.

			See A121602 for the case a(8) = 9 pertaining to A121603(8) = 11 unit circles.

E. Friedman, Erich's Packing Center: "Circles in Circles"

Cf. A121603, A121570, A121598, A121601, A121602.

A371858 Decimal expansion of Integral_{x=0..oo} 1 / (1 + x^7) dx.

Original entry on oeis.org

1, 0, 3, 4, 3, 7, 6, 0, 5, 5, 2, 6, 6, 7, 9, 6, 4, 8, 2, 9, 4, 5, 3, 0, 6, 4, 0, 6, 5, 1, 2, 4, 8, 8, 7, 4, 8, 3, 6, 4, 2, 5, 6, 7, 2, 6, 4, 2, 7, 3, 3, 7, 5, 8, 1, 0, 2, 8, 3, 3, 2, 6, 8, 8, 1, 5, 2, 5, 9, 3, 1, 0, 0, 7, 4, 8, 6, 2, 5, 4, 8, 5, 5, 5, 2, 0, 7, 5, 8, 9, 3, 8, 1, 8, 2, 0, 0, 0, 5, 9, 6, 0

Offset: 1

Ilya Gutkovskiy, Apr 09 2024

			1.0343760552667964829453064065124887483642567...

Index entries for transcendental numbers.

Decimal expansion of Integral_{x=0..oo} 1 / (1 + x^k) dx: A019669 (k=2), A248897 (k=3), A093954 (k=4), A352324 (k=5), A019670 (k=6), this sequence (k=7), A352125 (k=8).

Cf. A019674, A047336, A121598.

RealDigits[(1/7) Pi Csc[Pi/7], 10, 102][[1]]

Equals (1/7) * Pi * csc(Pi/7).
Equals A019674 * A121598.
Equals Product_{k>=2} (1 + (-1)^k/A047336(k)). - Amiram Eldar, Nov 22 2024

A367480 Decimal expansion of the radius of a common circle surrounded by seven tangent unit circles.

Original entry on oeis.org

1, 3, 0, 4, 7, 6, 4, 8, 7, 0, 9, 6, 2, 4, 8, 6, 5, 0, 5, 2, 4, 1, 1, 5, 0, 2, 2, 3, 5, 4, 6, 8, 5, 5, 1, 1, 3, 4, 4, 4, 5, 0, 1, 8, 8, 7, 6, 0, 6, 3, 2, 1, 1, 6, 2, 0, 6, 3, 1, 0, 6, 2, 9, 6, 4, 6, 6, 8, 5, 3, 3, 4, 2, 7, 7, 8, 4, 7, 9, 5, 9, 6, 3, 7, 9, 1, 1, 1, 4, 2, 1, 9, 7, 4, 7, 6, 1, 7, 9, 3, 6, 1, 5, 1, 5

Offset: 1

Thomas Otten, Dec 23 2023

The radius of a common circle surrounded by n tangent unit circles (n > 2) is r = 1/sin(Pi/n) - 1.

n=7 is the smallest number for which the radius cannot be expressed using square roots, since the regular heptagon formed by the centers of the tangent circles is non-constructible (see A246724, A188582, and A121570 for n=3, 4, 5).

			1.3047648709624865052...

Andrew M. Gleason, Angle Trisection, the Heptagon, and the Triskaidecagon, The American Mathematical Monthly 95, no. 3 (March 1988), pp. 185-194.
Index entries for algebraic numbers, degree 6.

Cf. A121598.

Cf. A121570, A188582, A246724.

RealDigits[Csc[Pi/7] - 1, 10, 120][[1]] (* Amiram Eldar, Dec 28 2023 *)

PARI
```
1/sin(Pi/7) - 1
```

Equals 1 / sin(Pi/7) - 1.
Equals A121598 - 1.
Largest of the 6 real-valued roots of 7*x^6+ 42*x^5 +49*x^4 -84*x^3 -119*x^2 +42*x-1=0. - R. J. Mathar, Aug 29 2025

More digits from Jon E. Schoenfield, Dec 24 2023

Comments edited by Michal Paulovic, Dec 26 2023

A384851 Decimal expansion of minimal radius of a circle that contains 14 non-overlapping unit disks.

Original entry on oeis.org

4, 3, 2, 8, 4, 2, 8, 5, 5, 4, 8, 6, 0, 8, 3, 6, 6, 8, 1, 4, 0, 3, 9, 0, 9, 3, 6, 7, 4, 7, 8, 1, 8, 1, 0, 9, 1, 6, 0, 8, 4, 9, 5, 7, 2, 9, 6, 5, 8, 6, 7, 5, 7, 0, 1, 2, 4, 5, 7, 5, 4, 8, 5, 5, 2, 2, 1, 1, 3, 3, 7, 0, 4, 5, 4, 0, 2, 1, 3, 8, 6, 3, 1, 9, 7, 5, 7

Offset: 1

Jinyuan Wang, Jun 10 2025

			4.328428554860836681403909367478181...

Dinesh B. Ekanayake and Douglas J. LaFountain, Tight partitions for packing circles in a circle, Italian Journal of Pure and Applied Mathematics, 51 (2024), 115-136.
Michael Goldberg, Packing of 14, 16, 17 and 20 Circles in a Circle, Mathematics Magazine, Vol. 44, No. 3 (May, 1971), pp. 134-139 (provides D/d=4.3284 in Table 1, page 136).

Cf. A084618, A121570, A121598, A121601, A121602, A281115, A282279.

1+sqrt(1+polrootsreal(Pol([707281, -27270266, 472689975, -4930771548, 34449512067, -168736166642, 591369611109, -1498751280720, 2767422383674, -3746579404052, 3734397946902, -2743990597288, 1486108072662, -593729401364, 175537055738, -38557290064, 6295485573, -759438450, 66647843, -4134492, 172311, -4346, 49]))[10])

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A121601 Decimal expansion of cosecant of 22.5 degrees = csc(Pi/8).

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A323601 Decimal expansion of sin(Pi/7).

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A374957 Decimal expansion of the circumradius of a regular heptagon with unit side length.

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A121603 Numbers n such that the radius of the smallest circle into which n unit circles can be packed is 1 + csc(Pi/k), where k >= 2 is an integer.

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A121604 Numbers k such that the radius of the smallest circle into which A121603(m) unit circles can be packed is 1 + csc(Pi/k).

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A371858 Decimal expansion of Integral_{x=0..oo} 1 / (1 + x^7) dx.

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A367480 Decimal expansion of the radius of a common circle surrounded by seven tangent unit circles.

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